Linearisation of finite abelian subgroups of the Cremona group of the plane

Details Linearisation of finite abelian subgroups of the Cremona group of the plane Linearisation of finite abelian subgroups of the Cremona group of the plane

2007-04-04

arxiv.org/abs/0704.0537v2

Groups Geom. Dyn. 3 (2009), no. 2, 215-266

Mathematics

Algebraic Geometry

41 pages, 2 figures, is a part of the author's PHD whose full text is available at math.AG/0610368. To appear in Groups Geom.

Scientific paper

10.4171/GGD/55

This article gives the proof of results announced in [J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris, S\'er. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces. Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to Z/2ZxZ/4Z, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles.

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